Introduction

In Computer Science, a binary tree is a hierarchical structure of nodes, each node referencing at most to two child nodes. Every binary tree has a root from which the first two child nodes originate. If a node has no children, then such nodes are usually termed leaves, and mark the extent of the tree structure.

A particular kind of binary tree, called the binary search tree, is very useful for storing data for rapid access, storage, and deletion. Data in a binary search tree are stored in tree nodes, and must have associated with them an ordinal value or key; these keys are used to structure the tree such that the value of a left child node is less than that of the parent node, and the value of a right child node is greater than that of the parent node. Sometimes, the key and datum are one and the same. Typical key values include simple integers or strings, the actual data for the key will depend on the application. In this article, I describe a binary search tree that stores string/double pairs. That is, the key is the string value and the data associated with the key is a double value. Developers can search the tree using string values.

Background

There are a number of basic operations one can apply to a binary search tree, the most obvious include, insertion, searching, and deletion.

To insert a new node into a tree, the following method can be used. We first start at the root of the tree, and compare the ordinal value of the root to the ordinal value of the node to be inserted. If the ordinal values are identical, then we have a duplicate and we return to the caller indicating so. If the ordinal value is less than the root, then we follow the left branch of the root, else we follow the right branch. We now start the comparison again but at the branch we took, comparing the ordinal value of the child with the node to be inserted. Traversal of the tree continues in this manner until we reach a left or right node which is empty and we can go no further. At this point, we insert the new node into this empty location. Note that new nodes are always inserted as leaves into the tree, and strictly speaking, nodes are thus appended rather than inserted.

Searching a binary search tree is almost identical to inserting a new node except that we stop the traversal when we find the node we're looking for (during an insertion, this would indicate a duplicate node in the tree). If the node is not located, then we report this to the caller.

Both insertion and searching are naturally recursive and are, arguably, easier to understand when considered in terms of their unit operation. A basic recursive search algorithm will look like:

In the source code provided with this article, insertion is implemented recursively, while searching uses an iterative approach.

Deletion is a little bit more complicated but boils down to three rules. The three rules refer to deleting nodes without any children, nodes with one child, and nodes with two children. If a node has no children, then the node is simply deleted. If the node has one child, then the node is deleted and the child node is brought forward to link to the parent. The complication occurs when a node has two children. However, even here, the rules are straightforward when stated. To delete a node with two children, the next ordinal node (called the successive node) on the right branch is used to replaced the deleted node. The successive node is then deleted. The successive node will always be the left most node on the right branch (likewise, the predecessor node will be the right most node on the left branch). The figure below illustrates the deletion rules.

A common alternative to using binary search tree is to use Hash tables. Hash tables have better search and insertion performance metrics. In theory, the time it takes to insert or search for an item in a Hash table is independent of the number of data items stored. In contrast, a binary search tree scales with log (N) where N is the number of data items (still far better than a linear search). The .NET libraries contain explicit support for Hash tables.

Balanced Trees

The time taken to insert or search for a specific item in a tree will be affected by a tree's depth. Deep trees take longer to search, and the insertion order into a tree can affect a tree's shape. A random insertion order will generally produce a more bushy and hence shallower tree compared to an ordered insert. Bushy trees are often called balanced trees, and although not implemented here, balancing a tree is a highly desirable feature for a binary search tree implementation. Certain algorithms such as the red-black tree will auto-balance as the tree is constructed (see Red/Black tree animation). The figure below shows three trees generated by three identical data sets but inserted in a different order. The first is the most balanced and hence the most shallow of the three examples.

Implementing the search and insertion methods using a recursive approach has the potential to yield poor performance, particularly when the trees are unbalanced.

Using the Code

Using the source code provided with this article is very easy. The following code illustrates the instantiation of a new binary tree, the insertion of data into the tree, and subsequent retrieval. The method insert() is used to insert new data, and the method findSymbol() is used to locate and retrieve data. If findSymbol() fails to locate the data item, it returns null. When successful, findSymbol() returns a TTreeNode object which has two properties, name and value. The following code illustrates how to use the binary search tree. The class name of the binary tree is TBinarySTree, and the individual nodes have class type TTreeNode.

delete will delete the node with the given key. If the method fails to locate the node, the method throws a simple exception.

The source code is licensed under the BSD license. The source should compile on C# 2.0.

To use the source code, unpack the source, load the binary tree solution (binaryTree.sln) and compile the BinaryTree project. In your own project, include as a reference to BinaryTree.dll. This version was written using Visual Studio 2003.

Points of Interest

The following graphs compare the performance of the Binary Tree search with .NET's built-in Hashtable class. The implementations follow the expected scaling laws as the number of stored data items increase. The X axis indicates the number of data items stored, ranging from 1000 to 1,000,000 items (21 intervals in all). The Y axis indicates the average time required to retrieve one data item (averaged over 20,000 retrieval attempts). The Hashtable follows roughly O(1), that is the time taken to retrieve data is independent of the number of data items stored. In contrast, the binary search tree scales roughly O(log (N)). However, this is far better than a linear search which would scale as O(N); that is, doubling the number of stored items doubles the average time taken to retrieve a single data item. The graph on the left shows the data plotted on log axes.

Times were computed using the QueryPerformanceCounter() method. The code for timing was derived from Tobi+C#=T#.

Other Possibilities

One should consider the implementation outlined here as the minimum practical implementation. The project where this implementation originated did not require any further sophistication. However, there are a number of areas where it could be significantly improved. In particular, two areas warrant further work:

The current implementation is specific to storing name/value pairs. Ideally, one would prefer a more generic implementation where a developer could employ their own object type.

The implementation may suffer performance degradation when subjected to large data sets if the trees become significantly unbalanced. Ideally one could implement the Red/Black variant to avoid this issue (see reference 1. at the end of the article for details).

Other minor changes include using a property in place of the public method count, adding further utility methods, and changing the naming convention on the classes and methods to make them more consistent with .NET.

History

Fixed a small error in the third tree, Figure 3 (missing C node).

There is an older article on CodeProject which discusses Red-Black trees in C#, something I should have spotted earlier (Red-Black Trees in C#).

References

There appears to be very little material on Binary Search Trees using .NET 1.1; the following, particularly the first link, provide material related to .NET 2.0.

You seem to have not accounted for the successor's possible right subtree when deleting a node with both children existing. You seem to be assuming that the successor would necessary be a leaf node, but couldn't it have a right subtree and still be a successor?

Or it is possible that I am missing something (since your article is very highly rated and I am a bit tired right now

It's been some time since I looked at the code so couldn't tell you off hand. I do use the code quite a bit and havnt yet observed any problems. Take another look, if the problem is really then we can fix it.

I recently found a need to use a BTree for storing large amounts of data (1-2 GB) and being able to quickly search through it. I tried a linked list but the search time was too slow for our needs. A binary tree was the best approach. Thanks to your code I have found a great solution. I'm storing 16 million items in the tree (1.5 GB) and random access times are under a ms, usually .005 - .008 ms. Your code was great with no bugs and with some slight tweaks for our specific object node is great. Awesome job!!!

There are 10 kinds of people in those world. Those who understand binary and those who don't.

You don't actually need to calculate the height of a Tree in order to balance a tree. Each node in an AVL Tree is either LeftHigh, Balanced or RightHigh - that's all you need. See http://BenedictBede.com for further information (the I# subsection of I++).

I have a few questions regarding the accuracy of the Count function in a couple of scenarios. The first being when a new node is called with insert. According to your insert code block (NOTE: variable names have changed for my purposes):

wouldn't you want to also add this node to the array list here so it has a count of one? Also, in your delete function although you set the value of the node to null, I believe the ArrayList still counts null values when you call the count function. Otherwise, nice piece of code!

You might be right. The reason why I wrote it was because I had some legacy code in Delphi that used such a function and I just translated it over to C#. If I didn't it again perhaps I would do something different.

My question is little funny. I had added implementation of red-black tree in this application, for comparing results. I got results for searching, e.g. number 10 graded -5. I am little confused with this number, it is hard to get string from tree whose structure has 5 000 000 string in this short time. Can you help me to explain this?

The algorithm is useful anywhere you need to store large amounts of data and need to access that data quickly. For example you might have a million user names plus their details and you want to quickly find a particular person in the list. Rather than search the list one item at a time you could use a binary tree to very rapidly locate the name.

Thanks for your answer but I don't see the binary nature of the problem even in this example. Don't misunderstand me, I really can't it is probably a limit of mines. I don't see ho to split a million username in parent and two childs an so on.

Maybe there are some articles on the net that can give me an introduction, ifyou know some, please point me there.

In order to store data in a binary tree you must first convert the data into some kind of numerical form. For example if you have a million unique user names, you can convert each name in to a number, this could be done by converting the letters in the names to the ascii numbers. Once in numeric form you can then decide whether a given name goes to the left or right side of a tree node depending on its value.

To make the problem simpler, let's say you only have five numbers, 8, 3, 10, 1, 6 and you want to store these in a binary tree. To start with the binary tree is empty. The first number is eight, and with this we create a node that represents the first node in the tree. The next number is a 3, this number is less than the first node, therefore we create a child node to the left of the parent node. The next number is a 10, this number is greater than the first node, therefore we create a child node to the right of the parent node. The next number is 1, since this is less than the first node, we go to the left, but the left node already has a child so we compare the number, 1 to the left child node. Since 1 is less than 3, we create a left node on the child to store the 1 and so on. The best way to learn this is to write out the process on paper and to build a picture of the tree yourself.

Check the section marked Background, this gives a discussion on how to insert information into the tree. Getting information out of the tree is the same but the reverse process.

"Insertion begins as a search would begin; if the root is not equal to the value, we search the left or right subtrees as before. Eventually, we will reach an external node and add the value as its right or left child, depending on the node's value. In other words, we examine the root and recursively insert the new node to the left subtree if the new value is less than the root, or the right subtree if the new value is greater than or equal to the root."

Now it's clearer. Suppose you have a Point p1 = new Point(3, 4) and a list with 1,000,000 points: you want to know if there is a point in the list equal to p1. Can we use this algorithm to avoid looping the all array elements and therefore saving time?

I have a problem that no seems to one could help with, I use the Chart Framework, ZedGraph (i also here i code project) and i try to handle a huge amount of data, lets say > 10 000 000 points!! Yes its huge, and now i have serious problem.

I have wondered if i should use some kind of hashtable for it, and store the index in a binary tree, but im not sure. I dont now how it could be used in this case, i have heard about it somewhere.Have you som idea to implement this in the case to to store the data?Im not so good in hashtable, and have short time to think of what strategy i should work on.

Binary trees can be difficult to work with if you're not used to them. I would recommend using a hashtable. Since C# has hashtable support that would be the easiest thing to do. Just google hashtable and C#.